can you elaborate more? what current density are you using and in what solvent? what is your potential window? are you sure that you don't have any metal that could cause a redox reaction? how do you prepare your electrodes?
I had some troubles like those using H2SO4 (1M), I was depositing my porous carbon on glassy carbon electrodes, and after drying water water was not wetting the surface of the electrode. The GCD curve for charging was OK, but discharging was not reaching the lover potential value. I just solve it by dipping the electrode on my solvent for some time before measuring. Hope this helps!
Sounds like your electrodes reached a plateau when at the upper potential limit. The reason could be the decomposition of the electrolyte you used during test.
An easy and straightforward way to check if the non-linearity is caused by the electrolyte decomposition is to lower the upper potential limit. If you see the linearity improved when reducing the upper limit, it probably due to the electrolyte decomposition.